Strain tensor selection and the elastic theory of incompatible thin sheets

Abstract

The existing theory of incompatible elastic sheets uses the deviation of the surface metric from a reference metric to define the strain tensor [Efrati et al., J. Mech. Phys. Solids 57, 762 (2009)]. For a class of simple axisymmetric problems we examine an alternative formulation, defining the strain based on deviations of distances (rather than distances squared) from their rest values. While the two formulations converge in the limit of small slopes and in the limit of an incompressible sheet, for other cases they are found not to be equivalent. The alternative formulation offers several features which are absent in the existing theory. (a) In the case of planar deformations of flat incompatible sheets, it yields linear, exactly solvable, equations of equilibrium. (b) When reduced to uniaxial (one-dimensional) deformations, it coincides with the theory of extensible elastica; in particular, for a uniaxially bent sheet it yields an unstrained cylindrical configuration. (c) It gives a simple criterion determining whether an isometric immersion of an incompatible sheet is at mechanical equilibrium with respect to normal forces. For a reference metric of constant positive Gaussian curvature, a spherical cap is found to satisfy this criterion except in an arbitrarily narrow boundary layer.

Modeling the formation of double rolls from heterogeneously patterned gels

“…This final configuration consists of two cylindrical rolls that are connected by a transition layer around the discontinuous line of swelling, z = L. (We note that this body of revolution differs from the one considered in Ref. [55] in the sense that its axis of revolution does not intersect with a point on the body. )…”

“…One is the work of internal forces to deform the lengths of in-plane line elements, and the second is the work of bending moments to deform the gel out of its initial flat position. This energy is given by [32,55],…”

“…To close the formulation, we need to relate the displacements u r and ζ to the strains. It was recently shown that the mathematical formulation of thin sheets that undergo axisymmetric deformations is considerably simplified if the 3D strain tensor is redefined [55]. Since the present problem falls under this axisymmetric category, we take advantage of this simplification and use Biot's strain [55][56][57] rather than the Green-St. Venant's strain [46,47] to formulate the elastic energy.…”

Contrasting bending energies from bulk elastic theories

“…Biot and similar measures have arisen explicitly or implicitly in both soft matter physics and continuum mechanical works, either as a natural consequence of bead-spring models of molecular mesostructures [22,23], from a desire to obtain simple constitutive relations between moment and kinematic variables [15,24], or from a recognition that a reduction process will generate certain direct theories of rods [25][26][27][28]. Such direct theories [29][30][31] for low-dimensional objects are constructed from a convenient choice of kinematic variables without consideration of whether they inherit their form from a bulk elastic theory.…”

Some observations on variational elasticity and its application to plates and membranes

“…Unlike Green and Almansi, we do not currently have a covariant expression for Biot and Swainger energies in terms of derivatives for the general case. Oshri and Diamant [81] have formulated a non-covariant two-dimensional theory for axisymmetrically deformed plates that employs Biot strains. I will not attempt here to formulate a general covariant three-dimensional description in terms of Biot or Swainger strain in 3D, but one can infer that the Swainger bending energy will be qualitatively similar to the Almansi description, in which extending an arc with fixed radius is a pure stretch, whereas increasing a circle's radius involves both stretching and bending.…”